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Amplitude models are constructed to describe the resonance structure of${D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}}$ and ${D^{0} \toK^{+}\pi^{-}\pi^{-}\pi^{+}}$ decays using $pp$ collision data collected atcentre-of-mass energies of 7 and 8 TeV with the LHCb experiment, correspondingto an integrated luminosity of $3.0\mathrm{fb}^{-1}$. The largest contributionsto both decay amplitudes are found to come from axial resonances, with decaymodes $D^{0} \to a_1(1260)^{+} K^{-}$ and $D^{0} \to K_1(1270/1400)^{+}\pi^{-}$ being prominent in ${D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}}$ and$D^{0}\to K^{+}\pi^{-}\pi^{-}\pi^{+}$, respectively. Precise measurements ofthe lineshape parameters and couplings of the $a_1(1260)^{+}$, $K_1(1270)^{-}$and $K(1460)^{-}$ resonances are made, and a quasi model-independent study ofthe $K(1460)^{-}$ resonance is performed. The coherence factor of the decays iscalculated from the amplitude models to be $R_{K3\pi} = 0.459\pm0.010\,(\mathrm{stat}) \pm 0.012\,(\mathrm{syst}) \pm 0.020\,(\mathrm{model})$,which is consistent with direct measurements. These models will be useful infuture measurements of the unitary-triangle angle $\gamma$ and studies of charmmixing and $C\!P$ violation.

Studies of the resonance structure in $D^{0} \to K^\mp \pi^\pm \pi^\pm \pi^\mp$ decays